So to buy the house from the estate, the estate signs the house to you, and you pay sister £137,500.

Obviously, any other assets are distributed by the estate on top of that, 50/50. If there's another £50k in assets, you may wish to forego your £25k worth, and offset that against the payment you make to your sister, so you pay her £112,500 for her 50% of the property.

No. She is keeping the 50k. Therefore she needs to access 25k to add to my half. So she borrows my half value of £162500 which ignores the 50k she has. She then borrows an extra 25k rather than ise the cash that she has. Therefore I get £187500 which is the same as her mortgage as the deposit is her half of the house of £167500.

Either the £50k was an unreserved gift - in which case, she doesn't need to "pay it back" to make your "inheritances level" - or it was a loan. It can't be both. Either way, because of the short timescale, it counts as part of the estate for IHT purposes - if it was longer, it'd tail off annually over 7 years, but <1yr counts as 100%.

She doesn't need to physically pay it back, just to metaphorically put a £50k IOU in the pot.

No. She is keeping the 50k. Therefore she needs to access 25k to add to my half. So she borrows my half value of £162500 which ignores the 50k she has. She then borrows an extra 25k rather than ise the cash that she has. Therefore I get £187500 which is the same as her mortgage as the deposit is her half of the house of £167500.

I can't follow your logic. If she borrows £162.5K plus an extra £25k to pay you £187.5K she will be left with £50k in her pocket?

Yes and half the house plus she will have bought the other half. We both end up with a value of £187500. If we sold externally, from the proceeds of the sale I would get £187500 and she would keep her 50k and her share of the house making her share £187500. It is because she getting a mortgage and choosing to keep the 50k that she has to borrow an extra 25k to give to me.

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